Abstract

Laser beams used for bar-code scanning exhibit speckle noise generated by the roughness of the surface on which bar codes are printed. Statistical properties of a photodetector signal that integrates a time-varying speckle pattern falling on its aperture are analyzed in detail. We derive simple closed-form expressions for the autocorrelation function and the power spectral density of the detector current for scanning beams with arbitrary field distributions. Theoretical calculations are illustrated by numerical simulations as well.

© 2001 Optical Society of America

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References

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  1. J. C. Dainty, ed., Laser Speckle and Related Phenomena, 2nd ed. (Springer-Verlag, Berlin, 1984).
  2. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 9–75.
  3. J. C. Dainty, “Recent developments,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 321–337.
  4. T. Okamoto, T. Asakura, “The statistics of dynamic speckles,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, Amsterdam, 1995), Vol. 39, pp. 185–248.
  5. D. Yu, M. Stern, J. Katz, “Speckle noise in laser bar-code-scanner system,” Appl. Opt. 35, 3687–3694 (1996).
    [CrossRef] [PubMed]
  6. G. F. Marshall, Optical Scanning (Marcel Dekker, New York, 1991).
  7. J. H. Churnside, H. T. Yura, “Velocity measurement using laser speckle statistics,” Appl. Opt. 20, 3539–3541 (1981).
    [CrossRef] [PubMed]
  8. L. I. Goldfisher, “Autocorrelation function and power spectral density of laser-produced speckle patterns,” J. Opt. Soc. Am. 55, 247–253 (1965).
    [CrossRef]
  9. V. V. Anisimov, S. M. Kozel, G. R. Lokshin, “Space–time statistical properties of coherent radiation scattered by a moving diffuse reflector,” Opt. Spectrosc. (USSR) 27, 258–262 (1969).
  10. T. Iwai, N. Takai, T. Asakura, “Dynamic statistical properties of laser speckle produced by a moving diffuse object under illumination of a Gaussian beam,” J. Opt. Soc. Am. 72, 460–467 (1982).
    [CrossRef]
  11. T. Yoshimura, “Statistical properties of dynamic speckles,” J. Opt. Soc. Am. A 3, 1032–1054 (1986).
    [CrossRef]
  12. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  13. G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 77–122.
  14. L. Bergstein, E. Marom, “Speckle pattern in polychromatic light” (manuscript available from the authors).
  15. A. E. Ennos, “Speckle interferometry,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 203–253.
  16. H. T. Yura, B. Rose, S. G. Hanson, “Dynamic laser speckle in complex ABCD optical systems,” J. Opt. Soc. Am. A 15, 1160–1166 (1998).
    [CrossRef]
  17. H. T. Yura, B. Rose, S. G. Hanson, “Speckle dynamics from in-plane rotating diffuse objects in complex ABCD optical systems,” J. Opt. Soc. Am. A 15, 1167–1173 (1998).
    [CrossRef]

1998 (2)

1996 (1)

1986 (1)

1982 (1)

1981 (1)

1969 (1)

V. V. Anisimov, S. M. Kozel, G. R. Lokshin, “Space–time statistical properties of coherent radiation scattered by a moving diffuse reflector,” Opt. Spectrosc. (USSR) 27, 258–262 (1969).

1965 (1)

Anisimov, V. V.

V. V. Anisimov, S. M. Kozel, G. R. Lokshin, “Space–time statistical properties of coherent radiation scattered by a moving diffuse reflector,” Opt. Spectrosc. (USSR) 27, 258–262 (1969).

Asakura, T.

T. Iwai, N. Takai, T. Asakura, “Dynamic statistical properties of laser speckle produced by a moving diffuse object under illumination of a Gaussian beam,” J. Opt. Soc. Am. 72, 460–467 (1982).
[CrossRef]

T. Okamoto, T. Asakura, “The statistics of dynamic speckles,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, Amsterdam, 1995), Vol. 39, pp. 185–248.

Bergstein, L.

L. Bergstein, E. Marom, “Speckle pattern in polychromatic light” (manuscript available from the authors).

Churnside, J. H.

Dainty, J. C.

J. C. Dainty, “Recent developments,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 321–337.

Ennos, A. E.

A. E. Ennos, “Speckle interferometry,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 203–253.

Goldfisher, L. I.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 9–75.

Hanson, S. G.

Iwai, T.

Katz, J.

Kozel, S. M.

V. V. Anisimov, S. M. Kozel, G. R. Lokshin, “Space–time statistical properties of coherent radiation scattered by a moving diffuse reflector,” Opt. Spectrosc. (USSR) 27, 258–262 (1969).

Lokshin, G. R.

V. V. Anisimov, S. M. Kozel, G. R. Lokshin, “Space–time statistical properties of coherent radiation scattered by a moving diffuse reflector,” Opt. Spectrosc. (USSR) 27, 258–262 (1969).

Marom, E.

L. Bergstein, E. Marom, “Speckle pattern in polychromatic light” (manuscript available from the authors).

Marshall, G. F.

G. F. Marshall, Optical Scanning (Marcel Dekker, New York, 1991).

Okamoto, T.

T. Okamoto, T. Asakura, “The statistics of dynamic speckles,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, Amsterdam, 1995), Vol. 39, pp. 185–248.

Parry, G.

G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 77–122.

Rose, B.

Stern, M.

Takai, N.

Yoshimura, T.

Yu, D.

Yura, H. T.

Appl. Opt. (2)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Opt. Spectrosc. (USSR) (1)

V. V. Anisimov, S. M. Kozel, G. R. Lokshin, “Space–time statistical properties of coherent radiation scattered by a moving diffuse reflector,” Opt. Spectrosc. (USSR) 27, 258–262 (1969).

Other (9)

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

G. Parry, “Speckle patterns in partially coherent light,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 77–122.

L. Bergstein, E. Marom, “Speckle pattern in polychromatic light” (manuscript available from the authors).

A. E. Ennos, “Speckle interferometry,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 203–253.

J. C. Dainty, ed., Laser Speckle and Related Phenomena, 2nd ed. (Springer-Verlag, Berlin, 1984).

J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 9–75.

J. C. Dainty, “Recent developments,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), pp. 321–337.

T. Okamoto, T. Asakura, “The statistics of dynamic speckles,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, Amsterdam, 1995), Vol. 39, pp. 185–248.

G. F. Marshall, Optical Scanning (Marcel Dekker, New York, 1991).

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Figures (6)

Fig. 1
Fig. 1

Free-space propagation model for investigating speckle. In most scanning applications the beam and the detector move in unison with respect to a fixed scattering surface.

Fig. 2
Fig. 2

Computer simulations of speckle patterns generated by a scanning Gaussian beam with λ=670 nm and radii ωx=ωy=0.12 mm. The separation between the scattering and observation planes is 5 cm, and each pixel equals 5.6 µm in the scattering plane. (a) Small beam displacement exhibits speckle translation mode, (b) large beam displacement exhibits speckle boiling mode.

Fig. 3
Fig. 3

Computer simulation of speckle noise resulting from integration of time-varying speckle patterns. The scattering surface is illuminated by a Gaussian beam with λ=670 nm and different radii ωx, ωy. The collecting aperture of area 1.5×1.5 mm is 5 cm from the scatterer. (a) ωx=ωy=0.06 mm, (b) ωx=ωy=0.12 mm.

Fig. 4
Fig. 4

Normalized autocorrelation function of detector current when the surface is illuminated by a Gaussian beam or a uniform illumination source.

Fig. 5
Fig. 5

Normalized power spectral density (PSD) of detector current when the surface is illuminated by a Gaussian beam or a uniform illumination source.

Fig. 6
Fig. 6

Lens in the optical path between the scattering [(ξ, η)] and observation [(x, y)] planes.

Tables (3)

Tables Icon

Table 1 Speckle Noise Power: Comparison of Theoretical Values with Numerical Simulations for a Gaussian Beam (2ωx=2ωy=0.12 mm)

Tables Icon

Table 2 Speckle Noise Power: Comparison of Theoretical Values with Numerical Simulations for a Gaussian Beam (2ωx=2ωy=0.24 mm)

Tables Icon

Table 3 Comparison of Speckle Noise Power for Two Circular Beams and One Elliptical Gaussian Beam

Equations (102)

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pI(I)=1Iexp-II,
σI2=I2-I2=I2.
I(x, y)=k=1NIk(x, y),
pN(I)=1(N-1)!NI0NIN-1 exp-NII0.
p(I; P)=1PI0exp-2I(1+P)I0-exp-2I(1-P)I0,
limP0p(I; P)=p2(I),limP1p(I; P)=p1(I).
ic=RA(x, y)Id(x, y)dxdy,
ic=RA(x, y)Id(x, y)dxdy=RIdS,
S=A(x, y)dxdy.
RId(x1, y1; x2, y2)=Id(x1, y1)Id(x2, y2),
RId(Δx, Δy)=Id2[1+|μ(Δx, Δy)|2],
μ(Δx, Δy)=|U(ξ, η)|2 expi2πλz(ξΔx+ηΔy)dξdη|U(ξ, η)|2dξdη,
σic2=R2A(x1, y1)A(x2, y2)×Id(x1, y1)Id(x2, y2)dx1dy1dx2dy2-ic2.
σic2=ic2M,
M-1=1S2RA(Δx, Δy)|μ(Δx, Δy)|2dΔxdΔy
RA(Δx, Δy)=A(x, y)A(x+Δx, y+Δy)dxdy.
M-11S2RA(0, 0)|μ(Δx, Δy)|2dΔxdΔy.
Sm=S2RA(0, 0)=A(x, y)dxdy2A2(x, y)dxdy,
Sc=|μ(Δx, Δy)|2dΔxdΔy.
M=SmSc,
Sc=(λz)2I2(ξ, η)dξdηI(ξ, η)dξdη2.
Id(x, y)=k=1NIk(x, y)
RN(x1, y1; x2, y2)=k=1NIk(x1, y1)Ik(x2, y2)+klIk(x1, y1)Il(x2, y2).
RN(Δx, Δy)=I021+1N|μ(Δx, Δy)|2,
σic2=ic2-ic2=ic2NM,
σic2=ic2ScNSm.
U(x, y)=U0 exp-x2ωx2-y2ωy2.
μ(Δx, Δy)=exp-12πλz2[(ωxΔx)2+(ωyΔy)2],
Sc=(λz)2πωxωy.
Dc=2πλzω1.27λz2ω.
Pd=ρPπz2S cos(θ).
Pd=A(x, y)Id(x, y)dxdy=SId.
Id=ρPπz2cos(θ).
ic(t)=RA(x, y)Id(x, y; t)dxdy,
ic(t)=RA(x, y)Id(x, y; t)dxdy=RSId.
ic(t)=RSρPπz2cos(θ)=RρΩdP.
Ric(t1, t2)=ic(t1)ic(t2)=R2A(x1, y1)A(x2, x2)×RId(x1, y1, x2, y2; t1t2)dx1dy1dx2dy2,
RId(x1, y1, x2, y2; t1, t2)=Id(x1, y1; t1)Id(x2, y2; t2)
R12(x1, y1, x2, y2)=Id1(x1, y1)Id2(x2, y2)
R12(Δx, Δy)=ρΩdS2P1P2[1+|μ12(Δx, Δy)|2],
μ12(Δx, Δy)=U1(ξ, η)U2*(ξ, η)expi2πλz(ξΔx+ηΔy)dξdη|U1(ξ, η)|2dξdη|U2(ξ, η)|2dξdη1/2.
μ(Δx, Δy;τ)=U(ξ, η)U*(ξ-Vτ, η)expi2πλz(ξΔx+ηΔy)dξdη|U(ξ, η)|2dξdη.
maxΔx,Δyμ(Δx, Δy;τ)=|μ(0, 0;τ)|=U0(ξ, η)U0(ξ-Vτ, η)dξdηU02(ξ, η)dξdη,
U(ξ, η)=U0(ξ, η)expiπλξ2Rx+η2Ry,
μ(Δx, Δy;τ)=exp-ik2Rx(Vτ)2|U0(ξ, η)|2dξdηU0(ξ, η)U0*(ξ-Vτ, η)×expi2πλzΔx+VzRxτξ+Δyηdξdη,
maxΔx,Δy|μ(Δx, Δy;τ)|=μ-VzRxτ, 0;τ=U0(ξ, η)U0(ξ-Vτ, η)dξdηU02(ξ, η)dξdη.
RId(Δx, Δy;τ)=ρΩdPS2[1+|μ(Δx, Δy;τ)|2].
Ric(τ)=RρΩdPS2×A(x, y)A(x+Δx, y+Δy)×[1+|μ(Δx, Δy;τ)|2]dxdydΔxdΔy.
Ric(τ)=ic21+1S2RA(Δx, Δy)×|μ(Δx, Δy;τ)|2dΔxdΔy.
Ric(τ)=ic21+1NS2RA(Δx, Δy)×|μ(Δx, Δy;τ)|2dΔxdΔy.
Ric(τ)=ic21+1NSm|μ(Δx, Δy;τ)|2dΔxdΔy.
|μ(Δx, Δy;τ)|2dΔxdΔy=(λz)2|U(ξ, η)|2|U(ξ-Vτ, η)|2dξdη|U(ξ, η)|2dξdη2,
Ric(τ)=ic21+(λz)2NSm×I(ξ, η)I(ξ-Vτ, η)dξdηI(ξ, η)dξdη2,
Ric(τ)=ic21+ScNSmI(ξ, η)I(ξ-Vτ, η)dξdηI2(ξ, η)dξdη.
Sic(f)=Ric(τ)exp(i2πfτ)dτ.
Sic(f)=ic2δ(f)+ScNSm×I(ξ, η)I(ξ-Vτ, η)exp(i2πfτ)dξdηdτI2(ξ, η)dξdη.
I(ξ, η)I(ξ-Vτ, η)exp(i2πfτ)dξdτ=1VI(ξ, η)expi2πfVξdξ2;
Sic(f)=ic2δ(f)+ScVNSm×I(ξ, η)expi2πfVξdξ2dηI2(ξ, η)dξdη.
Pic=Sic(f)df=Ric(0),
Pic=ic21+ScNSm.
PnPs=ScNSm=(λz)2NSmI2(ξ, η)dξdηI(ξ, η)dξdη2.
σic2=ic2ScNSm.
U(ξ, η)=2Pπωxωy1/2 exp-ξωx2+ηωy2×expiπλξ2Rx+η2Ry,
Sc=(λz)2πωxωy.
|μ(Δx, Δy; τ)|2=exp-Vτωx2×exp-πλz2ωx2Δx+VzRxτ2+(ωyΔy)2.
(ωxΔx)2+(ωyΔy)22π2(λz)2.
AG=2(λz)2πωxωy=2Sc.
Ric(τ)=ic21+ScNSmexp-Vτωx2.
Sic(f)=ic2δ(f)+πScNSmωxVexp-π2ωxfV2.
U(ξ, η)=γPωxωy1/2sincγξωxsincγηωy,
μ(Δx, Δy)=ΛωxγλzΔxΛωyγλzΔy,
Λ(x)=1-|x|if|x|<10otherwise.
Sc=|μ(Δx, Δy)|2dΔxdΔy=49γ2π(λz)2πωxωy=89(λz)2πωxωy,
AF=8π(λz)2ωxωy=9Sc.
Ric(τ)=ic21+6ScNSm×2πγ(Vτ/ωx)-sin[2πγ(Vτ/ωx)][2πγ(Vτ/ωx)]3.
limτ0Ric(τ)=ic21+ScNSm.
Sic(f)=ic2δ(f)+1γ32ScNSmωxVΛ21γωxfV.
RN(τ)=NSmωxωy(λz)2Ric(τ)ic2-1
SN(f)=NSmVωy(λz)2Sic(f)ic2-δ(f).
R12(x1, y1, x2, y2)=Id1(x1, y1)Id2(x2, y2).
Idl(x, y)=U1(ξ, η)Ul*(ξ¯, η¯)exp{i[ϕ(ξ, η)-ϕ(ξ¯, η¯)]}h(x-ξ, y-η)×h*(x-ξ¯, y-η¯)dξdηdξdη¯,
h(x, y)=exp(ikz)iλzexpik2z(x2+y2),
R12(x1, y1, x2, y2)=U1(ξ1, η1)U1*(ξ¯1, η¯1)U2(ξ2, η2)U2*(ξ¯2, η¯2)exp{i[ϕ(ξ1, η1)-ϕ(ξ¯1, η¯1)]}×exp{i[ϕ(ξ2, η2)-ϕ(ξ¯2, η¯2)]}h(x1-ξ1, y1-η1)×h*(x1-ξ¯1, y1-η¯1)h(x2-ξ2, y2-η2)h*(x2-ξ¯2, y2-η¯2)dξ1dη1dξ¯2dη¯2.
exp{i[ϕ(ξ1, η1)-ϕ(ξ¯1, η¯1)]}×exp{i[ϕ(ξ2, η2)-ϕ(ξ¯2, η¯2)]}=A(ξ1, η1; ξ¯1, η¯1)A(ξ2, η2; ξ¯2, η¯2)+A(ξ1, η1; ξ¯2, η¯2)A(ξ2, η2; ξ¯1, η¯1),
exp{i[ϕ(ξ, η)-ϕ(ξ¯, η¯)]}=Kδ(ξ-ξ¯)δ(η-η¯)
exp{i[ϕ(ξ1, η1)-ϕ(ξ¯1, η¯1)]}×exp{i[ϕ(ξ2, η2)-ϕ(ξ¯2, η¯2)]}=K2[δ(ξ1-ξ¯1)δ(η1-η¯1)δ(ξ2-ξ¯2)δ(η2-η¯2)+δ(ξ1-ξ¯2)δ(η1-η¯2)δ(ξ¯1-ξ2)δ(η¯1-η2)].
R12(x1, y1, x2, y2)=K2(λz)4|U1(ξ, η)|2dξdη|U2(ξ, η)|2dξdη+U1(ξ, η)U2*(ξ, η)×expi2πλz[ξ(x2-x1)+η(y2-y1)]dξdη2.
μ12(Δx, Δy)=U1(ξ, η)U2*(ξ, η)expi2πλz(ξΔx+ηΔy)dξdη|U1(ξ, η)|2dξdη|U2(ξ, η)|2dξdη1/2,
R12(Δx, Δy)=K2(λz)4P1P2[1+|μ12(Δx, Δy)|2],
Idl=K(λz)2Pl.
Idl=ρΩdSPl,
R12(Δx, Δy)=ρΩdS2P1P2[1+|μ12(Δx, Δy)|2].
Ul(p, q)=Ul(ξ, η)exp[iϕ(ξ, η)]×h(p-ξ, q-η; l1)dξdη,l=1, 2,
h(p, q; s)=exp(iks)iλsexpik2s(p2+q2).
Ul(x, y)=Ul(p, q)exp-ik2 f(p2+q2)×h(x-p, y-q; l2)dpdq,l=1, 2.
Ul(x, y)=Ul(ξ, η)exp[iϕ(ξ, η)]hL(x, ξ)hL(y, η)dξdη,  l=1, 2,
hL(u, v)=bl1l2expiπλv2l11-bl1-2bl1l2uv+u2l21-bl2,
μ12(Δx, Δy)=U1(ξ, η)U2*(ξ, η)expi2πλbl1l2(ξΔx+ηΔy)dξdη|U1(ξ, η)|2dξdη|U2(ξ, η)|2dξ dη1/2.
μ(Δx, Δy;τ)=U(ξ, η)U*(ξ-Vτ, η)expi2πλbl1l2(ξΔx+ηΔy)dξdη|U(ξ, η)|2dξdη,
ze=l1+l2-l1l2f.
f<l1l22(l1+l2).
f>l1l22(l1+l2).

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